lol...i always used my "me no speak english" or " english is not my first language" BS when it happened. i always get away with it ha ha ha.
but yeah, i guess it really depend on the officer. one time i even gave my passport together with the GC, and the officer immediately return my passport and said " i dont need that, just this" while waving on my GC.
im on the waiting list for nexus approval. once it got approved, i can use global service kiosk too. less human interaction the better